Characters in Low-Dimensional Topology

该奖项支持美国的初级研究人员参加2018年6月2日至6日在加拿大蒙特利尔魁北克大学举行的“低维拓扑中的字符”会议，该会议由数学研究中心主办。会议为传播和讨论低维拓扑和几何的最新重要进展提供了一个国际论坛。这是纯数学中一个庞大而活跃的研究领域，其基本目标是理解被称为流形的数学对象的结构，流形在许多科学现象的建模中自然出现。流形是一种“空间”，其小尺度结构仅取决于自由度的数量，即维度：例如，在小尺度上，一维流形看起来就像一条线，二维流形看起来就像一个平面，三维流形看起来就像我们生活的空间区域，等等。虽然给定维度的所有流形在小尺度上都是相同的，但它们的大尺度结构可能非常不同且相当复杂；为了研究这个问题，人们开发了许多数学技术。“低维”指的是3维和4维，这是我们特别感兴趣的：如果我们只考虑空间宇宙，我们生活在一个具有三维的流形中，但如果包括时间，则是四维的。会议汇集了专家和新兴研究人员，报告最近的结果并探索未来的方向。该奖项允许美国的年轻数学家、研究生和博士后研究人员参加。将特别努力支持多样性，鼓励妇女和其他传统上在数学领域代表性不足的群体参加。会议的重点是许多不同的几何和拓扑技术，最近已开发的研究三维流形。目前已知3-流形具有刚性几何结构，最近的相关发展不仅为低维拓扑以及邻近领域带来了新的见解。同时，起源于量子物理和规范理论的新同调理论，在结理论和三维和四维拓扑等老问题上取得了巨大进展。尽管在几个方面取得了这些进展，但相对而言，人们对3维分析、几何和代数不变量之间的相互作用知之甚少。会议强调了正在进行的研究，重点是寻找三维拓扑的这几个方面之间的直接联系。会议汇集了研究3流形的新不变量、应用于3流形的几何群论以及更经典的几何和拓扑技术的专家和早期职业数学家。该奖项反映了美国国家科学基金会的法定使命，并通过基金会的智力价值和更广泛的影响审查标准进行了评估，认为值得支持。

This award supports participation by U.S.-based junior researchers in the conference "Characters in Low-Dimensional Topology" held June 2-6, 2018 in Montreal, Canada at the University of Quebec at Montreal, hosted by the Centre de Recherches Mathematiques. The conference provides an international forum for the dissemination and discussion of recent important advances in low-dimensional topology and geometry. This is a large and active area of research in pure mathematics, whose basic goal is to understand the structure of mathematical objects known as manifolds, which arise naturally in modeling many scientific phenomena. Manifolds are "spaces" whose small-scale structure depends only on the number of degrees of freedom, called the dimension: for example, on a small scale a 1-dimensional manifold looks just like a line, a 2-dimensional manifold looks just like a flat plane, a 3-dimensional manifold looks like a region of the space in which we live, and so on. Although all manifolds of a given dimension are the same on a small scale, their large-scale structures can be very different and quite complicated; many mathematical techniques have been developed to investigate this. "Low-dimensional" means dimensions 3 and 4, which are of special interest: we live in a manifold that has three dimensions if we consider only the spatial universe, but four dimensions if time is included as well. The conference brings together experts and emerging researchers to report on recent results and explore future directions. This award enables young mathematicians from the U.S., graduate students and postdoctoral researchers, to participate. A particular effort will be made to support diversity by encouraging the attendance of women and other groups that are traditionally underrepresented in mathematics.The conference focuses on the many different geometric and topological techniques that have recently been developed to study 3-dimensional manifolds. It is now known that 3-manifolds possess rigid geometric structures, and recent related developments have brought new insights not only to low-dimensional topology but also to neighboring fields. Meanwhile, new homology theories, having their origins both in quantum physics and in gauge theory, have led to dramatic progress on old questions in knot theory and 3-and 4-dimensional topology. Despite these advances on several fronts, relatively little is known about the interplay between analysis, geometry, and algebraic invariants in dimension 3. The conference highlights ongoing research focused on finding direct connections between these several aspects of 3-dimensional topology. The meeting brings together experts and early-career mathematicians working on the new invariants of 3-manifolds, on geometric group theory applied to 3-manifolds, and on more classical geometric and topological techniques. The website for the conference is cirget.uqam.ca/boyerfest//enThis award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.